The proof can be found in Appendix REF . We use an upper bound on the regret of general OMD from Bubeck [1]} and relate it to our setting; in doing so, we bound the diameter of \(\mathcal {X}\)  w.r.t. Bregman divergence under \(\Phi \)  as well as the dual-norm \(\left\Vert \cdot \right\Vert _p\)  of the gradients \(\nabla f_{\mathbf {r}_t}(\mathbf {x}_t)\) .
